Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDual Lindstedt series and KAM theorem
118
0
0.0
(
0
)
Authors
Marco Frasca
Ask ChatGPT about the research
No Arabic abstract
We prove that exists a Lindstedt series that holds when a Hamiltonian is driven by a perturbation going to infinity. This series appears to be dual to a standard Lindstedt series as it can be obtained by interchanging the role of the perturbation and the unperturbed system. The existence of this dual series implies that a dual KAM theorem holds and, when a leading order Hamiltonian exists that is non degenerate, the effect of tori reforming can be observed with a system passing from regular motion to fully developed chaos and back to regular motion with the reappearance of invariant tori. We apply these results to a perturbed harmonic oscillator proving numerically the appearance of tori reforming. Tori reforming appears as an effect limiting chaotic behavior to a finite range of parameter space of some Hamiltonian systems. Dual KAM theorem, as proved here, applies when the perturbation, combined with a kinetic term, provides again an integrable system.
rate research
Read More
We prove an analytic KAM-Theorem, which is used in [1], where the differential part of KAM-theory is discussed. Related theorems on analytic KAM-theory exist in the literature (e. g., among many others, [7], [8], [13]). The aim of the theorem presented here is to provide exactly the estimates needed in [1].
The KAM iterative scheme turns out to be effective in many problems arising in perturbation theory. I propose an abstract version of the KAM theorem to gather these different results.
We prove a Gannon-Lee theorem for non-globally hyperbolic Lo-rentzian metrics of regularity $C^1$, the most general regularity class currently available in the context of the classical singularity theorems. Along the way we also prove that any maximizing causal curve in a $C^1$-spacetime is a geodesic and hence of $C^2$-regularity.
We show that the Hawking--Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of $C^{1, 1}$-regularity. We formulate appropriate wea
Heun differential equations are the most general second order Fuchsian equations with four regular singularities. An explicit integral series representation of Heun functions involving only elementary integrands has hitherto been unknown and noted as an important open problem in a recent review. We provide explicit integral representations of the solutions of all equations of the Heun class: general, confluent, bi-confluent, doubly-confluent and triconfluent, with integrals involving only rational functions and exponential integrands. All the series are illustrated with concrete examples of use. These results stem from the technique of path-sums, which we use to evaluate the path-ordered exponential of a variable matrix chosen specifically to yield Heun functions. We demonstrate the utility of the integral series by providing the first representation of the solution to the Teukolsky radial equation governing the metric perturbations of rotating black holes that is convergent everywhere from the black hole horizon up to spatial infinity.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
